Network Connectivity and Systematic Risk - Monica Billio. December, 15 2014. CSRA research meeting.

1. Network Connectivity,
Systematic Risk and
Diversification
SYstemic Risk TOmography:
Signals, Measurements, Transmission Channels, and
Policy Interventions
Monica Billio, University Ca' Foscari Venezia (Italy)
Massimiliano Caporin, University of Padova (Italy)
Roberto Panzica Goethe University Frankfurt (Germany)
Loriana Pelizzon University Ca' Foscari Venezia (Italy) and Goethe
University Frankfurt (Germany)
CSRA research meeting – December, 15 2014

2. Network Connectivity, Systematic Risk and
Diversification
Monica Billio1
Massimiliano Caporin2
Roberto Panzica3
Loriana Pelizzon1,3
1
University Ca’ Foscari Venezia (Italy)
2
University of Padova (Italy)
3
Goethe University Frankfurt (Germany)
CSRA, December, 2014
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 1 / 18

3. Research questions
Introduction
There is a general agreement on the traditional decomposition of an asset
(portfolio) total risk into systematic and idiosyncratic components following
the CAPM model
Systematic risks comes from the dependence of returns on common factors
Idiosyncratic risks are asset-specific
But...
There is also a recent consensus on the existence of systemic risks
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 2 / 18

4. Research questions
Introduction
Systemic risk definition: any set of circumstances that threatens the stability
of, or public confidence in, the financial system
Systemic risk is a function of a system
Systemic risk arises endogenously from a system
Systemic risk is a function of connections between and the structure of
financial institutions or, more generally, between the companies and/or
economic sectors
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 3 / 18

5. Research questions
(Challenging) Research questions
What is the impact of network connectivity on the asset return loading
to common systematic factors?
Does network connectivity endanger the power of diversification?
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 4 / 18

6. Literature review
Measuring systemic links
An increasing literature in economics investigates the role of interconnections
between different firms and sectors, functioning as a potential propagation
mechanism of idiosyncratic shocks throughout the economy.
Canonical idea: Lucas (1977), among others, that states that such
microeconomic shocks would average out and thus, would only have
negligible aggregate effects. Similarly, these shocks would have little impact
on asset prices.
However:
Acemoglou et al. (2011) use network structure to show the possibility that
aggregate fluctuations may originate from microeconomic shocks to firms.
Such a possibility is discarded in standard macroeconomics models due to a
“diversification argument”.
Shock propagation in static networks Horvath (1998, 2000), Dupor (1999),
Shea (2002), and Acemoglu, Carvalho, Ozdaglar, and Tahbaz-Salehi (2011).
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 5 / 18

7. Literature review
Pricing and diversification in multifactor models
General multi-factor model for a k−dimensional vector of risky assets (m risk
factors)
Rt = E [Rt] + BFt + t (1)
Under equilibrium expected returns depend on the factor risk premiums Λ
E [Rt] = rf + BΛ (2)
Standard total risk decomposition
VAR [Rt] = BVAR [Ft] B + VAR [ t] (3)
ΣR = BΣF B + Ω (4)
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 6 / 18

8. Literature review
Pricing and diversification in multifactor models
Portfolio risk decomposition (ω being a k−dimensional vector of portfolio
weights)
VAR [ω Rt] = ω BVAR [Ft] B ω + ω VAR [ t] ω (5)
ω ΣR ω = ω BΣF B ω + ω Ωω (6)
Diversification benefit
limk→∞ω Ωω = ν (7)
Special case: uncorrelated idiosyncratic shocks with average variance ¯σ2
limk→∞ω Ωω =
1
k
¯σ2
= 0 (8)
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 7 / 18

9. The model
An augmented multifactor model with network connectivity
Network connectivity represents contemporaneous relations across assets that
co-exists with the dependence on common systematic risk factors
Network connections are contemporaneous relations across endogenous
variables
A (Rt − E [Rt]) = BFt + t (9)
The simultaneous equation system above is not identified unless we impose
some restriction; k number of assets is much larger than m the number of
factors
Assumption 1: the idiosyncratic shocks are uncorrelated, that is Ω is a
diagonal matrix
Assumption already taken into account in multi-factor models
Key idea: Predetermined networks provide information on the existence of
links across assets and on the strength/intensity of the link
Networks can be represented as spatial matrices
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 8 / 18

10. The model
A general framework with systemic and systematic risks
By means of spatial matrices we can impose a structure on matrix A and
rewrite the simultaneous equation system
A = I − ρW (10)
(Rt − E [Rt]) = ρW (Rt − E [Rt]) + BFt + t (11)
The coefficient ρ represents the impact coming from neighbours (by now we
assume it is a scalar) and W represents neighbour relationships
This simultaneous equation system corresponds to a Spatial Auto Regression
Panel model where the covariates (risk factors) are common across all
subjects (at least in a simplified representation)
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 9 / 18

11. The model
Our contributions
Take a multifactor model and focus on both pricing and diversification effects
Measure systemic links starting from a network capturing causality relations
Show that network links act as an inflating factor on the asset loadings to the
common factors
Show that network elements impact on both the systematic and idiosyncratic
risk components
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 10 / 18

12. The model
Network connectivity impact on expected returns
We therefore have:
(I − ρW ) (Rt − E [Rt]) = BFt + t (12)
It holds that
(I − ρW )
−1
= I + ρW + ρ2
W 2
+ ρ3
W 3
. . . (13)
Therefore the model corresponds to
Rt = E [Rt] + BFt +
∞
j=1
ρj
W j
BFt + ηt (14)
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 11 / 18

13. The model
Network connectivity impact on expected returns
Under equilibrium the expected returns equal
E [Rt] = rf + BΛ +
∞
j=1
ρj
W j
BΛ (15)
where Λ represents systematic factor risk premia
Therefore, as ρ > 0 and elements in W are positive, the presence of
network/systemic links inflates the loading to the factors with an impact on
the asset expected returns
Expected returns increase as a consequence to an increase in ρ or a change in
W with subsequent effects on prices
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 12 / 18

14. The model
Network connectivity impact on risk
We can play around this decomposition to recover a more insightful one
VAR [Rt] = A−1
BΣF B A−1
+ A−1
Ω A−1
(16)
= ¯BΣF
¯B + AΩA (17)
= ¯BΣF
¯B + AΩA ± BΣF B ± Ω (18)
= BΣF B
i
+ Ω
ii
+ ¯BΣF
¯B − BΣF B
iii
+ (AΩA − Ω)
iv
(19)
We have thus four terms in the risk decomposition
i The structural systematic component
ii The structural idiosyncratic component
iii The network connectivity impact on the structural systematic component
iv The network connectivity impact on the idiosyncratic component
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 13 / 18

15. The model
Systemic links impact on risk
This has effects on the diversification benefits which can be analytically
derived in a special case
Consider K uncorrelated idiosyncratic shocks with average variance ¯σ2
and a
W matrix where all assets are linked to each other, we have
limK→∞ω AΩηA ω =
K + ρ2
(K + ρ)
2
(ρ − 1)
2 ¯σ2
= 0 (20)
Diversification benefits still present but the decrease of the idiosyncratic
component of the portfolio variance is much slower
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 14 / 18

16. The model
Systemic links impact on risk
Portfolio idiosyncratic risk across different ρ levels and increasing number of
assets. The case ρ = 0 corresponds to the absence of spatial links and is the
standard result for diversification benefits.
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 15 / 18

17. The model
Simulated examples
1/N portfolio variance decomposition across different values of ρ with the
same ”random” matrix W (relative decomposition)
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 16 / 18

18. The model
Future developments
Generalize the model in four directions
First by adding heterogeneity to the asset reaction to the systemic links
A = I − RW (21)
R = diag (ρ1, ρ2, . . . , ρN ) (22)
Second, by allowing for a time-variation in the W matrix
A ⇒ At = I − RWg(t) (23)
In this way the matrices Wg(t) capture the dynamic in the links across assets
(the evolution of the newtork), while the spatial coefficients included in R
represent the impact on each asset of the neighbours, and are assumed to be
time-independent
The Wg(t) matrices are also assumed to be known at time t and thus we
condition the model estimation and the analysis to their availability
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 17 / 18

19. The model
Future developments
We assume the evolution of Wg(t) is smooth or acts at a time scale much
lower than that affecting the evolution of asset returns (i.e. for monthly
returns the W change with, say, a yearly frequency); to highlight this aspect
we index the spatial links matrices to a function of time
Deeper evaluation of risk factor exposure and pricing implications; analysis of
diversification impact of network exposure
Third: time-variation in the ρ coefficients
Fourth: different design of the W matrices taking into account the strength
of the relation across subjects
Billio, Caporin, Panzica and Pelizzon N&S December, 2014 18 / 18

20. This project has received funding from the European Union’s
Seventh Framework Programme for research, technological
development and demonstration under grant agreement n° 320270
www.syrtoproject.eu
This document reflects only the author’s views.
The European Union is not liable for any use that may be made of the information contained therein.